D ec 2 00 6 The Lie Algebra of the Group of Bisections ⊳ A Chapter in Synthetic Differential Geometry of Groupoids ⊲

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avantgarde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framework of synthetic differential geometry that the tangent space of the group of bisections of a microlinear groupoid at its identity is naturally a Lie algebra. We give essentially distinct two proofs for its Jacobi identity.